Well if you actually have the time then time isn't really that much of a factor... cutting the openings and endings and previews, doing research ahead of time so you can cut through the filler arcs should you wish, at about 20 mins an episode, 3 episodes an hour, a full 26 episode anime can be finished within 10 hours... a 52 episode series within a standard day still allowing for 8 hours for sleep, light exercise and cleaning up. Considering a vast majority of anime comes in either series of 12 or 24 episodes, you should be able to munch through a vast variety in a week, consuming a significant percentage within a month... and as stated earlier in regards to cutting the filler arcs, if you can push sleeping off here and there (a "just one more episode" mentality), a series as vast as One Piece can devoured in a matter of days.

Here's some advice for those wishing to attempt... also helpful just for anime binging on series (like one piece and similar).

Reduce sleeping time to about 4-6 hours at a time, but remember to keep schedule, (if you put off sleep too long too frequently, it WILL catch up potentially more than doubling the time you remain unconscious without warning).

Choose meals that can be consumed easily either by hand and/or at a desk with minimal effort, (having to make use of a knife can be distracting especially when watching subtitled shows). Easily clear-able foods are an important factor, keep food that makes use of crockery too a minimum. Also avoid ramen or soupy noodles as they can make a mess and cause wasted time on cleaning.

Allocate a few hours to showering and light exercise, taking a walk in the mornings should have you feeling refreshed and ready the day to come.

Make sure your computer is capable of handling what you're about to do! unless you intend to take from streaming and DVD all the anime you are about to watch, (which will be difficult especially when dealing with older and unlicensed series), you are going to be doing a lot of downloading. Use at least a duel core processor and potentially a second (SATA) hard drive (not USB, with the exception of storage only), this way you can watch uninterrupted video whilst downloading in the background, and since you're bound to be downloading Giga-bites worth of data, the second hard drive helps as it wont weigh down the main hard drive on which your OS is installed which can slow your computer.

for those in the UK and countries where Broadband can be limited make sure you have a suitable broadband package for the task, 40GB per month is a stretch but it is manageable, but it is recommended you get the Unlimited packages as these usually come with less restrictive throttling among some of the service providers.

Back to advice for all, in regards to internet usage, for the period of your anime binge, go Nocturnal. this will reduce the problems of annoyingly slow connections especially when streaming, since there are fewer people online (at least locally). This may not counter server traffic issues, but ISP loads should be lower off peak and they are less likely to throttle connections during the late nights early mornings.

And last and probably most helpful tip, get a Megavideo premium membership (or at least one paid membership to one of the commonly used file upload sites). This will be helpful both when streaming but also when dealing with older series downloads since it's joint with Megaupload. When on this venture you will find that torrents just wont get you through, Direct Downloads are going to be your only source at times and file upload sites can be very restricting when mass downloading, and basically you will hit a lot of walls when going at it for free. It REALLY is worth the cost getting the premium option.

Barriers you will come across in attaining your goal...

Language. Not everything has be subtitled or translated. Various shows get picked up but make a bad impression with the community that they are dropped shortly after (even shows that managed to get high enough status in japan that a second series of 50+ episodes was made, Gokujou!! Mecha Mote Iinchou is an example of this [dropped after 2 episodes])... you can find raws of these shows though so if you know your Japanese you should be fine.

pre DVD anime, attempts have been made to make a large number of these shows available on DVD over time, however this wont happen for all the licensed series that have been released over the years. VHS options may still be avaliable of course. There has been an attempt by some uploaders to get these series online, being possibly 2nd generation VHS uploads though using hard subs, and TV NTSC resolution, the quality is terrible and often unreadable after the encoding process, (try not to destroy your eyes decrypting the subs on these videos.

Other than that, most unlicensed and out of production series are still available for download if you look hard enough (good luck with this as older less popular anime DVDs are hard to find even second hand and not ridiculously overpriced at that, Saint Tail [one of Tokyopop's attempt at anime] is an example of this... look hard enough though and you can get it in High Definition soft subs).

once you sort out time and finances, a dream of watching all the anime in the world really is within your grasp.

sure if you have time, money, can watch the newest as they happen, buy the oldest ones and collect every version there is in every medium there is and since this is every anime ever that includes every language and raws

even If I could do this I wouldn't, there are some animes/genera's that I'm not very fond of

it's probably going to take a while, but i just think that it will take you away from what you reallyyy should be doing.
i'm already a loser watching a lot of anime that my grades are lowering at school and i barely eat 3 meals a day o_o
i become lazy and... pretty much unable to do anything but watch anime.

I'm starting to watch almost 20 anime series at the same time. if i go more than that, i don't think i'll get my face off the computer.
if you watch every anime possibly made, i think it will damage your health.. unless you don't watch them all at the same time like i do.

I think you can do it!! New seasons happen but that's 1 episode per week, you could catch up in the meantime!! I wanna watch soooo much anime I see the world as anime xD I sometimes see people with subtitles when they speak

Did a very low-ball estimate of how long it would take:
Assumptions:
1) every anime series is 13 episodes long
2) You only watch anime tv series that myanimelist.net knows about: so that's 2316 series
3) Every episode is about 23 minutes long

crunching the number gets you 13*23*2316 = 692484 minutes. So if you made it your job to watch anime and spent 8 hours a day watching it, it'd take almost 4 years. Or "only" 1.3 years if you did nothing but watch anime.

This is a very low estimate since it doesn't include the really long series like Naruto, Bleach, etc., it also doesn't include anime movies or ovas. So, it might be doable, it'd just take a long time to do.